Restore the mass m of the electron and show that at high energy, s →∞, the integrated cross sect

step 1 The energy of the electron is given by EE equals HF minus HF prime plus MEC squared and the mome

Restore the mass m of the electron and show that at high...

Restore the mass $m$ of the electron and show that at high energy, $s \rightarrow \infty$, the integrated cross section for Compton scattering is $$ \sigma=\frac{1}{64 \pi^{2} s} \int \overline{\left.|T|^{2}\right|^{2}} d \Omega \rightarrow \frac{2 \pi \alpha^{2}}{s} \log \left(\frac{s}{m^{2}}\right) $$ Note that at high energy the dominant contribution comes from $9 \mathrm{R}_{2}$, via a glancing collision in which the $u$-channel electron is almost on mass shell.

Restore the mass $m$ of the electron and show that at high energy, $s \rightarrow \infty$, the integrated cross section for Compton scattering is
$$
\sigma=\frac{1}{64 \pi^{2} s} \int \overline{\left.|T|^{2}\right|^{2}} d \Omega \rightarrow \frac{2 \pi \alpha^{2}}{s} \log \left(\frac{s}{m^{2}}\right)
$$
Note that at high energy the dominant contribution comes from $9 \mathrm{R}_{2}$, via a glancing collision in which the $u$-channel electron is almost on mass shell.

Key Concepts

Integrated Cross Section and Phase Space Integration

The integrated cross section is obtained by averaging the squared scattering amplitude over the allowed phase space of the final state particles, typically involving an angular integration over the solid angle. This integration can reveal logarithmic enhancements and dependencies on parameters like s and the electron mass, which are central to understanding the behavior of the process at high energies.

Mandelstam Variables

Mandelstam variables, including s, t, and u, are Lorentz-invariant quantities used to describe the kinematics of particle scattering. The variable s represents the square of the total center-of-mass energy and plays a crucial role in analyzing the high energy behavior of scattering amplitudes, particularly in identifying which contributions dominate in various limits.

Dominance of Specific Scattering Channels

In many high energy scattering processes, certain kinematic channels can dominate due to the particular behavior of the propagators in the Feynman diagrams. For instance, when an intermediate state in the u-channel is nearly on-shell, it leads to an enhanced contribution, often manifesting as logarithmic terms in the integrated cross section. This emphasizes the significance of specific diagrams and the corresponding energy regime in the overall scattering process.

Compton Scattering

Compton scattering is a fundamental quantum electrodynamics process in which a photon scatters off an electron, resulting in a redistribution of energy and momentum between the two particles. The process is described using Feynman diagrams and involves the exchange of virtual particles, capturing both tree-level and higher order effects in perturbation theory.

High Energy Limit

At high energies, the center-of-mass energy squared (denoted by the Mandelstam variable s) becomes much larger than the electron mass squared. In this regime, the mass of the electron can be restored explicitly in the formulas, yet its contribution is negligible in certain kinematic regions, leading to simplified expressions for observable quantities such as the integrated cross section.

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